Bounded Arithmetic in Free Logic Yoriyuki Yamagata CTFM, 2013/02/20 - - PowerPoint PPT Presentation

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Bounded Arithmetic in Free Logic Yoriyuki Yamagata CTFM, 2013/02/20 - - PowerPoint PPT Presentation

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Bounded Arithmetic in Free Logic Yoriyuki Yamagata CTFM, 2013/02/20 Busss theories 2 Language of Peano Arithmetic + # a # b = 2 | | BASIC axioms PIND , , ( ) 2

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Bounded Arithmetic in Free Logic

Yoriyuki Yamagata CTFM, 2013/02/20

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Buss’s theories 𝑇2

𝑗

  • Language of Peano Arithmetic + β€œ#”

– a # b = 2 𝑏 β‹…|𝑐|

  • BASIC axioms
  • PIND

𝐡 𝑦 2 , Ξ“ β†’ Ξ”, 𝐡(𝑦) 𝐡 0 , Ξ“ β†’ Ξ”, 𝐡(𝑒) where 𝐡 𝑦 ∈ Σ𝑗

𝑐, i.e. has 𝑗-alternations of

bounded quantifiers βˆ€π‘¦ ≀ 𝑒, βˆƒπ‘¦ ≀ 𝑒.

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PH and Buss’s theories 𝑇2

𝑗 𝑇2

1 = 𝑇2 2 = 𝑇2 3 = …

Implies 𝑄 = β–‘(𝑂𝑄) = β–‘(Ξ£2

π‘ž) = …

We can approach (non) collapse of PH from (non) collapse of hierarchy of Buss’s theories

(PH = Polynomial Hierarchy)

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Our approach

  • Separate 𝑇2

𝑗 by GΓΆdel incompleteness theorem

  • Use analogy of separation of 𝐽Σ𝑗
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Separation of 𝐽Σ𝑗

𝐽Σ3 ⊒ Con(IΞ£2) 𝐽Σ2 ⊒ Con IΞ£2 … 𝐽Σ1 βŠ† βŠ†

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Consistency proof inside 𝑇2

𝑗

  • Bounded Arithmetics generally are not

capable to prove consistency.

– 𝑇2 does not prove consistency of Q (Paris, Wilkie) – 𝑇2 does not prove bounded consistency of 𝑇2

1 (PudlΓ‘k)

– 𝑇2

𝑗 does not prove consistency the 𝐢𝑗 𝑐 fragement

  • f 𝑇2

βˆ’1 (Buss and IgnjatoviΔ‡)

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Buss and Ignjatović(1995)

… βŠ† 𝑇2

3 ⊒ 𝐢3 b βˆ’ Con(𝑇2 βˆ’1)

𝑇2

2 ⊒ 𝐢2 b βˆ’ Con(𝑇2 βˆ’1)

𝑇2

1 ⊒ 𝐢1 b βˆ’ Con(𝑇2 βˆ’1)

βŠ†

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SLIDE 8

Where…

  • 𝐢𝑗

𝑐 βˆ’ 𝐷𝐷𝐷 π‘ˆ

– consistency of 𝐢𝑗

𝑐 βˆ’proofs

– 𝐢𝑗

𝑐 βˆ’proofs : the proofs by 𝐢𝑗 𝑐-formule

– 𝐢𝑗

𝑐:Ξ£0 𝑐(Σ𝑗 𝑐)… Formulas generated from Σ𝑗 𝑐 by

Boolean connectives and sharply bounded quantifiers.

  • 𝑇2

βˆ’1

– Induction free fragment of 𝑇2

𝑗

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If…

𝑇2

π‘˜ ⊒ 𝐢i b βˆ’ Con 𝑇2 βˆ’1 , j > i

Then, Buss’s hierarchy does not collapse.

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Consistency proof of 𝑇2

βˆ’1 inside 𝑇2 𝑗

Problem

  • No truth definition, because
  • No valuation of terms, because
  • The values of terms increase exponentially
  • E.g. 2#2#2#2#2#...#2

In 𝑇2

𝑗 world, terms do not have values a priori.

  • Thus, we must prove the existence of values in proofs.
  • We introduce the predicate 𝐹 which signifies existence of

values.

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Our result(2012)

… βŠ† 𝑇2

5 ⊒ 3 βˆ’ Con(𝑇2 βˆ’1𝐹)

𝑇2

4 ⊒ 2 βˆ’ Con(𝑇2 βˆ’1𝐹)

𝑇2

3 ⊒ 1 βˆ’ Con(𝑇2 βˆ’1𝐹)

βŠ†

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Where…

  • 𝑗 βˆ’ 𝐷𝐷𝐷 π‘ˆ

– consistency of 𝑗-normal proofs – 𝑗-normal proofs : the proofs by 𝑗-normal formulas – 𝑗-normal formulas: Formulas in the form: βˆƒπ‘¦1 ≀ 𝑒1βˆ€π‘¦2 ≀ 𝑒2 … 𝑅𝑦𝑗 ≀ 𝑒𝑗𝑅𝑦𝑗+1 ≀ 𝑒𝑗+1 . 𝐡(… ) Where 𝐡 is quantifier free

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Where…

  • 𝑇2

βˆ’1𝐹

– Induction free fragment of 𝑇2

𝑗𝐹

– have predicate 𝐹 which signifies existence of values

  • Such logic is called Free logic
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𝑇2

𝑗𝐹(Language)

Predicates

  • =, ≀, 𝐹

Function symbols

  • Finite number of polynomial functions

Formulas

  • Atomic formula, negated atomic formula
  • 𝐡 ∨ 𝐢, 𝐡 ∧ 𝐢
  • Bounded quantifiers
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𝑇2

𝑗𝐹(Axioms)

  • 𝐹-axioms
  • Equality axioms
  • Data axioms
  • Defining axioms
  • Auxiliary axioms
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Idea behind axioms…

β†’ 𝑏 = 𝑏

Because there is no guarantee of 𝐹𝑏 Thus, we add 𝐹𝑏 in the antecedent

𝐹𝑏 β†’ 𝑏 = 𝑏

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E-axioms

  • 𝐹𝐹 𝑏1, … , π‘π‘œ β†’ πΉπ‘π‘˜
  • 𝑏1 = 𝑏2 β†’ πΉπ‘π‘˜
  • 𝑏1 β‰  𝑏2 β†’ πΉπ‘π‘˜
  • 𝑏1 ≀ 𝑏2 β†’ πΉπ‘π‘˜
  • ¬𝑏1≀ 𝑏2 β†’ πΉπ‘π‘˜
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Equality axioms

  • 𝐹𝑏 β†’ 𝑏 = 𝑏
  • 𝐹𝐹 𝑏

βƒ— , 𝑏 βƒ— = 𝑐 β†’ 𝐹 𝑏 βƒ— = 𝐹 𝑐

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Data axioms

  • β†’ 𝐹𝐹
  • 𝐹𝑏 β†’ 𝐹𝑑0𝑏
  • 𝐹𝑏 β†’ 𝐹𝑑1𝑏
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Defining axioms

𝐹 𝑣 𝑏1 , 𝑏2, … , π‘π‘œ = 𝑒(𝑏1, … , π‘π‘œ) 𝐹𝑏1, … , πΉπ‘π‘œ, 𝐹𝑒 𝑏1, … , π‘π‘œ β†’ 𝐹 𝑣 𝑏1 , 𝑏2, … , π‘π‘œ = 𝑒(𝑏1, … , π‘π‘œ)

𝑣 𝑏 = 0, 𝑏, 𝑑0𝑏, 𝑑1𝑏

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Auxiliary axioms

𝑏 = 𝑐 βŠƒ 𝑏#𝑑 = 𝑐#𝑑 𝐹𝑏#𝑑, 𝐹𝑐#𝑑, 𝑏 = |𝑐| β†’ 𝑏#𝑑 = 𝑐#𝑑

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PIND-rule

where 𝐡 is an Σ𝑗

𝑐-formulas

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Bootstrapping 𝑇2

𝑗𝐹 I. 𝑇2

𝑗𝐹 ⊒ Tot(𝐹) for any 𝐹, 𝑗 β‰₯ 0

  • II. 𝑇2

𝑗𝐹 ⊒ BASICβˆ—, equality axioms βˆ—

  • III. 𝑇2

𝑗𝐹 ⊒ predicate logic βˆ—

  • IV. 𝑇2

𝑗𝐹 ⊒ Σ𝑗 𝑐 βˆ’PINDβˆ—

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Theorem (Consistency)

𝑇2

𝑗+2 ⊒ i βˆ’ Con(𝑇2 βˆ’1𝐹)

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Valuation trees

a#a+b=19 a#a=16 b=3 a=2 ρ-valuation tree bounded by 19

ρ(a)=2, ρ(b)=3

𝑀 𝑏#𝑏 + 𝑐 , 𝜍 ↓19 19 𝑀 𝑒 , 𝜍 ↓𝑣 𝑑 is Ξ£1

𝑐

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Bounded truth definition (1)

  • π‘ˆ 𝑣, 𝑒1 = 𝑒2 , 𝜍 ⇔def

βˆƒπ‘‘ ≀ 𝑣, 𝑀 𝑒1 , 𝜍 ↓𝑣 𝑑 ∧ 𝑀 𝑒1 , 𝜍 ↓𝑣 𝑑

  • π‘ˆ 𝑣, 𝜚1 ∧ 𝜚2 , 𝜍 ⇔def

π‘ˆ 𝑣, 𝜚1 , 𝜍 ∧ π‘ˆ 𝑣, 𝜚2 , 𝜍

  • π‘ˆ 𝑣, 𝜚1 ∨ 𝜚2 , 𝜍 ⇔def

π‘ˆ 𝑣, 𝜚1 , 𝜍 ∨ π‘ˆ 𝑣, 𝜚2 , 𝜍

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Bounded truth definition (2)

  • π‘ˆ 𝑣, βˆƒπ‘¦ ≀ 𝑒, 𝜚(𝑦) , 𝜍 ⇔def

βˆƒπ‘‘ ≀ 𝑣, 𝑀 𝑒 , 𝜍 ↓𝑣 𝑑 ∧ βˆƒπ‘’ ≀ 𝑑, π‘ˆ 𝑣, 𝜚 𝑦 , 𝜍 𝑦 ↦ 𝑒

  • π‘ˆ 𝑣, βˆ€π‘¦ ≀ 𝑒, 𝜚(𝑦) , 𝜍 ⇔def

βˆƒπ‘‘ ≀ 𝑣, 𝑀 𝑒 , 𝜍 ↓𝑣 𝑑 ∧ βˆ€π‘’ ≀ 𝑑, π‘ˆ(𝑣, 𝜚 𝑦 , 𝜍[𝑦 ↦ 𝑒]) Remark: If 𝜚 is Σ𝑗

𝑐, π‘ˆ 𝑣, 𝜚 is Σ𝑗+1 𝑐

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induction hypothesis

𝑣: enough large integer 𝑠: node of a proof of 0=1 Γ𝑠 β†’ Δ𝑠: the sequent of node 𝑠 𝜍: assignment 𝜍 𝑏 ≀ 𝑣

βˆ€π‘£β€² ≀ 𝑣 βŠ– 𝑠, { βˆ€π΅ ∈ Ξ“

𝑠 π‘ˆ 𝑣′, 𝐡 , 𝜍

βŠƒ [βˆƒπΆ ∈ Ξ”r, π‘ˆ(𝑣′ βŠ• 𝑠, 𝐢 , 𝜍)]}

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Conjecture

  • 𝑇2

βˆ’1𝐹 is weak enough

– 𝑇2

𝑗+2 can prove 𝑗-consistency of 𝑇2 βˆ’1𝐹

  • While 𝑇2

βˆ’1𝐹 is strong enough

– 𝑇2

𝑗𝐹 can interpret 𝑇2 𝑗

  • Conjecture

𝑇2

βˆ’1𝐹 is a good candidate to separate 𝑇2 𝑗 and 𝑇2 𝑗+2.

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Future works

  • Long-term goal

𝑇2

𝑗 ⊒ π‘—βˆ’Con(𝑇2 βˆ’1𝐹)?

  • Short-term goal

– Simplify 𝑇2

𝑗𝐹

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Publications

  • Bounded Arithmetic in Free Logic

Logical Methods in Computer Science Volume 8, Issue 3, Aug. 10, 2012